Doorbell wire is 20awg, but thermostat wire is 18awg. Doorbell wire comes in pairs and thermostat wire comes in a variety of configurations including a pair, many odd numbered parings but also an 8 wire version.
If you took the 8-wire version, used every other wire to each of the two poles (4 wires per pole, pos and neg) then you would have an aggregate 12awg cable. The skin effect brigade will like the doorbell wire better. With that, I would use four of the twisted pairs, themselves twisted in a star-quad configuration so also 4 wires per pole, for an aggregate 14awg. Arguably, this would be the better cable of the two.
With either, use two separate runs for bi-wire. If you can find the wire with polyethylene insulation, that would be much better, IMO, than PE and even better than teflon. You could do much worse.
If you took the 8-wire version, used every other wire to each of the two poles (4 wires per pole, pos and neg) then you would have an aggregate 12awg cable. The skin effect brigade will like the doorbell wire better. With that, I would use four of the twisted pairs, themselves twisted in a star-quad configuration so also 4 wires per pole, for an aggregate 14awg. Arguably, this would be the better cable of the two.
With either, use two separate runs for bi-wire. If you can find the wire with polyethylene insulation, that would be much better, IMO, than PE and even better than teflon. You could do much worse.